Multiple positive solutions for the fractional Schr\"{o}dinger-Poisson systems involving critical nonlinearities with potential

TL;DR

This paper proves the existence of multiple positive solutions for fractional Schrödinger-Poisson systems with critical nonlinearities and sign-changing potentials, using variational methods and topological category theory.

Contribution

It introduces a new relationship between the number of solutions and the topological category of the potential's maximum set, extending previous results.

Findings

Multiple positive solutions are established for the system.

The number of solutions depends on the category of the maximum set of g(x).

A novel link between solution count and topological properties is demonstrated.

Abstract

In this paper, we study the existence of multiple positive solutions for a class of fractional Schr\"{o}dinger-Poisson systems involving sign-changing potential and critical nonlinearities on an unbounded domain. With the help of Nehari manifold and Ljusternik-Schnirelmann category, we investigate how the coefficient $g(x)$ of the critical nonlinearity affects the number of positive solutions. Moreover, we present a novel relationship between the number of positive solutions and the category of the global maximum set of $g(x)$.